Integrand size = 23, antiderivative size = 204 \[ \int \frac {1}{\left (c-d x^2\right ) \sqrt [3]{c+3 d x^2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} \sqrt {d} x}{\sqrt {c}}\right )}{2\ 2^{2/3} \sqrt {3} c^{5/6} \sqrt {d}}+\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {d} x}{\sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{c+3 d x^2}\right )}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}}-\frac {\text {arctanh}\left (\frac {\sqrt {c}}{\sqrt {d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{c+3 d x^2}\right )}{\sqrt {d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}} \]
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Time = 0.04 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {401} \[ \int \frac {1}{\left (c-d x^2\right ) \sqrt [3]{c+3 d x^2}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {d} x}{\sqrt [6]{c} \left (\sqrt [3]{2} \sqrt [3]{c+3 d x^2}+\sqrt [3]{c}\right )}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt {3} \sqrt {d} x}{\sqrt {c}}\right )}{2\ 2^{2/3} \sqrt {3} c^{5/6} \sqrt {d}}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{c+3 d x^2}\right )}{\sqrt {d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}}-\frac {\text {arctanh}\left (\frac {\sqrt {c}}{\sqrt {d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}} \]
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Rule 401
Rubi steps \begin{align*} \text {integral}& = -\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {d} x}{\sqrt {c}}\right )}{2\ 2^{2/3} \sqrt {3} c^{5/6} \sqrt {d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {d} x}{\sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{c+3 d x^2}\right )}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c}}{\sqrt {d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{c+3 d x^2}\right )}{\sqrt {d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt {d}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 4.94 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (c-d x^2\right ) \sqrt [3]{c+3 d x^2}} \, dx=\frac {3 c x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {3 d x^2}{c},\frac {d x^2}{c}\right )}{\left (c-d x^2\right ) \sqrt [3]{c+3 d x^2} \left (3 c \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {3 d x^2}{c},\frac {d x^2}{c}\right )+2 d x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},-\frac {3 d x^2}{c},\frac {d x^2}{c}\right )-\operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},-\frac {3 d x^2}{c},\frac {d x^2}{c}\right )\right )\right )} \]
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\[\int \frac {1}{\left (-d \,x^{2}+c \right ) \left (3 d \,x^{2}+c \right )^{\frac {1}{3}}}d x\]
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Timed out. \[ \int \frac {1}{\left (c-d x^2\right ) \sqrt [3]{c+3 d x^2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\left (c-d x^2\right ) \sqrt [3]{c+3 d x^2}} \, dx=- \int \frac {1}{- c \sqrt [3]{c + 3 d x^{2}} + d x^{2} \sqrt [3]{c + 3 d x^{2}}}\, dx \]
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\[ \int \frac {1}{\left (c-d x^2\right ) \sqrt [3]{c+3 d x^2}} \, dx=\int { -\frac {1}{{\left (3 \, d x^{2} + c\right )}^{\frac {1}{3}} {\left (d x^{2} - c\right )}} \,d x } \]
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\[ \int \frac {1}{\left (c-d x^2\right ) \sqrt [3]{c+3 d x^2}} \, dx=\int { -\frac {1}{{\left (3 \, d x^{2} + c\right )}^{\frac {1}{3}} {\left (d x^{2} - c\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (c-d x^2\right ) \sqrt [3]{c+3 d x^2}} \, dx=\int \frac {1}{\left (c-d\,x^2\right )\,{\left (3\,d\,x^2+c\right )}^{1/3}} \,d x \]
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